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We show that the Cheeger constant for $n$-dimensional isotropic logconcave measures is $O(n^{1/4})$, improving on the previous best bound of $O(n^{1/3}\sqrt{\log n}).$ As corollaries we obtain the same improved bound on the thin-shell estimate, Poincar\'{e} constant and Lipschitz concentration constant and an alternative proof of this bound for the isotropic (slicing) constant; it also follows that the ball walk for sampling from an isotropic logconcave density in ${\bf R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start.
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